If it was only the headphone jack that broke, you should still be able to connect to the unit with a data cable and pull the information off. Do not use the SYNC function or apple software, it doesn't always work properly!
I do NOT use apple software for my sync / upload / download. I utilize Media Monkey & have NO problems pulling all of my data off the Ipod & loading it into other units. I cloned mine with another for a x-mas present to the g/f. She still has half of the unit memory for additional songs and I had no problems with the above software. Also put the media monkey software on her computer so she could change things around as appropriate.
While I have no affiliation with their products, I do like how well it worked for me. For the price to go pro for life ($19); it was worth it in my mind. I also helped a few co-workers out who were having problems with their units & data syncing. The apple stuff wasn't working for them, but my program did.
HTH & Good Luck!